- Make a sketch of the function arising from the convolution of the two functions depicted in Fig. P.11.34.Fig. P.11.34.
- Use the method illustrated in Fig. 11.30 to convolve the two functions depicted in Fig. P.11.23. Fig. P.11.23.Fig. 11.30
- Consider the periodic function defined over one wavelength byDetermine the Fourier series representation of ƒ(x). Draw a diagram of ƒ(x).
- Suppose we have a single slit along the y-direction of width b where the aperture function is constant across it at a value of A0. What is the diffracted field if we now apodize the slit with a
- Show that F{ƒ(x) cos k0x} = [F(k - k0) + F(k + k0)]/2 and that F{ƒ(x) sin k0x} = [F(k - k0) - F(k + k0)]/2i.
- What is the linear separation between two identical points on an object that can just be resolved, if that object is at the near-point of the eye (25 cm). See the previous question.
- Using Rayleigh’s criterion, determine the smallest angle subtended by two points of equal brightness that can just be resolved by the human eye. Assume a pupil diameter of 2.0 mm and a mean
- We intend to observe two distant equal-brightness stars whose angular separation is 50.0 × 10-7 rad. Assuming a mean wavelength of 550 nm, what is the smallest-diameter objective lens that will
- Suppose that we have a laser emitting a diffraction-limited beam (λ0 = 632.84 nm) with a 2-mm diameter. How big a light spot would be produced on the surface of the Moon a distance of 376 × 103 km
- Consider the periodic functionE(t) = E0 cos ωtand suppose all of the negative half-cycles are removed. Determine the Fourier series representation of the resulting modified (“rectified”)
- Referring to the previous two problems with the cosine grating oriented horizontally, make a sketch of the electric-field amplitude along y' with no filtering. Plot the corresponding image irradiance
- Return to Eq. (12.21) and separate it into two terms representing a coherent and an incoherent contribution, the first arising from the superposition of two coherent waves with irradiances of
- Figure P.11.49 shows a transparent ring on an otherwise opaque mask. Make a rough sketch of its autocorrelation function, taking l to be the center-to-center separation against which you plot that
- Figure P.11.36 shows, in one dimension, the electric field across an illuminated aperture consisting of several opaque bars forming a grating. Considering it to be created by taking the product of a
- Figure P.11.30 shows two functions. Convolve them graphically and draw a plot of the result. Figure P.11.30 f(x) h(x) 3 1 1 2 3 4 1 2 3
- Prove that δ(x - x0) * ƒ(x) = ƒ(X - x0) and discuss the meaning of this result. Make a sketch of two appropriate functions and convolve them. Be sure to use an asymmetrical ƒ(x).
- Suppose we have two functions, ƒ(x, y) and h(x, y), where both have a value of 1 over a square region in the xy-plane and are zero everywhere else (Fig. P.11.21). If g(X, Y) is their convolution,
- Figure P.11.10 shows two periodic functions, ?(x) and h(x), which are to be added to produce g(x). Sketch g(x); then draw diagrams of the real and imaginary frequency spectra, as well as the
- Make a rough sketch of the irradiance function for a Fresnel diffraction pattern arising from a double slit. What would the Cornu spiral picture look like at point-P0?
- Use the Cornu spiral to make a rough sketch of |B̅12(w)|2 versus (w1 + w2)/2 for Δw = 5.5. Compare your results with those ofFig. 10.79. B2(w) 5000 4000 3000 2000 Aw = 2.5 Aw = 3.5 1000 100 μη -3
- Figure P.10.39 shows several aperture configurations. Roughly sketch the Fraunhofer patterns for each. Note that the circular regions should generate Airy-like ring systems centered at the origin.
- From symmetry considerations, create a rough sketch of the Fraunhofer diffraction patterns of an equilateral triangular aperture and an aperture in the form of a plus sign.
- Linear light polarized horizontally passes through a quarterwave plate whose fast axis is π/8 rad above the horizontal. Use the phasor method to graphically determine the polarization state of the
- Linear light oscillating along the x-axis is passed through a quarter-wave plate whose fast axis is 45° above the x-axis. Use the phasor method to graphically show that the emerging light is
- Draw a quartz Wollaston prism, showing all pertinent rays and their polarization states.
- Supposing that Fig. P.6.32 is to be imaged by a lens system suffering spherical aberration only, make a sketch of the image. Fig. P.6.32
- Please establish that the separation between principal planes for a thick lass lens is roughly one-third its thickness. The simplest geometry occurs with a planar-convex lens tracing a ray from the
- A nearsighted person with the same vision in both eyes has a far point at 100 cm and a near point at 18 cm, each measured from her cornea.(a) Determine the focal length of the needed corrective
- Draw a ray diagram locating the images of a point source as formed by a pair of mirrors at 90? (Fig. P.5.61a). Now create a ray diagram locating the images of the arrow shown in Fig. P.5.61b. Fig
- Consider a thin positive lens L1, and using a ray diagram, show that if a second lens L2 is placed at the focal point of L1, the magnification does not change. That’s a good reason to wear
- Redraw the ray diagram for a compound microscope (Fig. 5.110), but this time treat the intermediate image as if it were a real object. This approach should be a bit simpler.Fig. 5.110 Exit pupil fe
- The electric field of an electromagnetic plane wave is given in SI units by (a) What is the wave?s angular frequency? (b) Write an expression for vector k. (c) What is the value of k? (d) Determine
- Show, from the integral definitions, that ƒ(x) * g(x) = ƒ(x) * g(-x), where the functions are real.
- Figure P.11.44 depicts a single €œsaw tooth€ function and its autocorrelation. Explain why cÆ’Æ’ (X) is symmetrical about the origin. Why does it extend
- Prove analytically that the convolution of any function ƒ(x) with a delta function, δ(x), generates the original function ƒ(X).
- Examine the three graphs in Fig. P. 11.20 and explain what€™s being illustrated. Discuss how the shape of g(X) arises. Why is g(X) symmetrical about X = 0? What€™s the
- Prove that the area under the convolution of the functions ƒ(x) and h(x) equals the product of the areas under each of those functions.
- Prove that ƒ * h = h * ƒ directly. Now do it using the convolution theorem.
- Given F{ƒ(x)}, show that F{ƒ(x - x0)} differs from it only by a linear phase factor.
- Utilizing Eq. (11.38), show that F-1 {F{(x)}} = Æ’(x). о ik(x — x') дk 6(х — х') (11.38) 2т
- With the last two problems in mind, show that F{(1/2Ï€) Ã— sinc (1/2 x)} = rect(Îº), starting with the knowledge that F{rect(x)} = sinc (1/2 Îº), in other
- The rectangular function is often defined aswhere it is set equal to 1/2 at the discontinuities (Fig. P.11.14). Determine the Fourier transform ofNotice that this is just a rectangular pulse, like
- Show that the Fourier transform of the transform, F{ƒ(x)}, equals 2πƒ(-x), and that this is not the inverse transform of the transform, which equals ƒ(x). This problem was suggested by Mr. D.
- Given that F{ƒ(x)} = F(κ), introduce a constant scaling factor 1/α and determine the Fourier transform of ƒ(x/α). Show that the transform of ƒ(-x) is F(-κ).
- Compute the Fourier transform of the triangular pulse shown in Fig. P.11.11. Make a sketch of your answer, labeling all the pertinent values on the curve.Fig. P.11.11 ↑ f(x) -L +L
- A collimated beam from a ruby laser (694.3 nm) having an irradiance of 10 W/m2 is incident perpendicularly on an opaque screen containing a square hole 5.0 mm on a side. Compute the irradiance at a
- A horizontal hole 2.00 mm by 1.00 mm in an opaque screen is illuminated normally by a beam of collimated light of wavelength 500 nm. If the incident irradiance is 30.0 W/m2, calculate the approximate
- We want to make a Fresnel zone plate with a principal focal length of 2.00 m for krypton ion laser light of wavelength 647 nm. How big should the central transparent disk be? If it has 30 transparent
- Consider a Fresnel zone plate having a transparent circular disk at its center. This is the m = 1 region, and the tenth transparent region has a diameter of 6.00 mm. Determine the plate’s principal
- Considering the previous problem, suppose we insert an opaque disk of radius RD at the center of the hole so that the unobstructed region is now an annulus. If RD = 0.50 mm, determine the ratio of
- Envision an opaque screen (Σ) containing a circular hole of radius R. A point source S lies on the central axis a distance ρ0 in front of Σ and an observation point-P lies a distance
- Plane waves (λ = 550 nm) impinge normally on a 5.00-mm diameter hole in an opaque screen (Σ). The diffraction pattern is observed on another screen (σ), which is slowly moved toward the aperture.
- Imagine that you are looking through a piece of square woven cloth at a point source (Î»0 = 600 nm) 20 m away. If you see a square arrangement of bright spots located about the point
- Light having a frequency of 4.0 × 1014 Hz is incident on a grating formed with 10 000 lines per centimeter. What is the highest order spectrum that can be seen with this device? Explain.
- A beam of collimated polychromatic light ranging from 500 nm to 700 nm impinges normally on a transmission grating having 590 000 lines/m. If the complete second-order spectrum is to appear, how
- With Example 10.9 on page 494 in mind, determine the number of grooves a transmission grating must have if it is to resolve the sodium doublet in the first-order spectrum. Compare the results of both
- Light from a laboratory sodium lamp has two strong yellow components at 589.592 3 nm and 588.995 3 nm. How far apart in the first-order spectrum will these two lines be on a screen 1.00 m from a
- Collimated red light (656.281 6 nm) from a hydrogen discharge lamp falls perpendicularly onto a transmission grating. The beam emerges forming a red line in the second-order spectrum at an angle of
- A diffraction grating produces a second-order spectrum of yellow light (λ0 = 550 nm) at 25°. Determine the spacing between the lines on the grating.
- A diffraction grating with slits 0.60 × 10-3 cm apart is illuminated by light with a wavelength of 500 nm. At what angle will the third-order maximum appear?
- The Hubble Space Telescope has an objective mirror 2.4 m in diameter. With an average wavelength of 550 nm, determine its linear limit of resolution at 600 km (about 370 miles).
- How were blue light–emitting lasers used to improve DVD technology? Explain.
- The neoimpressionist painter Georges Seurat was a member of the pointillist school. His paintings consist of an enormous number of closely spaced small dots (≈ 1/10 inch) of pure pigment. The
- Figure P.12.26 shows two quasimonochromatic point sources of incoherent light illuminating two pinholes in a mask. Show that the fringes formed on the plane of observation have minimum visibility
- Suppose that we have a quasimonochromatic, uniform thermal slit source of incoherent light, such as a discharge lamp with a mask and filter in front of it. We wish to illuminate a region on an
- We wish to construct a double-pinhole setup illuminated by a uniform, quasimonochromatic, thermal slit source of incoherent light of mean wavelength 500 nm and width b, a distance of 1.5 m from the
- Return to Fig. 12.8 and the broad quasimonochromatic slit source. How wide should this source be if the fringe visibility is to be 0.9? The source is 1.0 m in front of the aperture screen and
- Return to Fig. 12.8 and the broad, quasimonochromatic, long, rectangular source of light (λ̅0 = 500 nm). How far apart should the two movable narrow aperture slits be if the fringe pattern on
- Suppose that Young’s double-slit apparatus is illuminated by sunlight with a mean wavelength of 550 nm. Determine the separation of the slits that would cause the fringes to vanish.
- Figure P.11.46 depicts a function Æ’(x). Draw, to scale, its autocorrelation function cÆ’Æ’(X). How wide is cÆ’Æ’(X)? How wide is each
- A rectangular pulse extends from -x0 to +x0 and has a height of 1.0. Sketch its autocorrelation, cƒƒ (X). How wide is cƒƒ (X)? Is it an even or odd function? Where does it start (become
- Consider the periodic functionÆ’(x) = cos (Îºx + Îµ)where the amplitude is 1.0, and P is an arbitrary phase term. Show that the autocorrelation function (before
- Graphically find the cross-correlation cÆ’h(x) of the two functions shown here:Figure P.11.41How wide will it be? At what value of x will the correlation peak? What is the maximum value of
- Show that when ƒ(t) = A sin (ωt + ε), Cƒƒ (τ) = (A2/2) cos ωt, which confirms the loss of phase information in the autocorrelation.
- Suppose a given aperture produces a Fraunhofer field pattern E(Y, Z). Show that if the aperture€™s dimensions are altered such that the aperture function goes from A(y, z) to
- Show (for normally incident plane waves) that if an aperture has a center of symmetry (i.e., if the aperture function is even), then the diffracted field in the Fraunhofer case also possesses a
- Imagine that we have Young’s Experiment, where one of the two pinholes is now covered by a neutral-density filter that cuts the irradiance by a factor of 10, and the other hole is covered by a
- Figure P.11.35 depicts a rect function (as defined above) and a periodic comb function. Convolve the two to get g(x). Now sketch the transform of each of these functions against spatial frequency
- Given the function ƒ(x) = δ(x + 3) + δ(x - 2) + δ(x - 5), convolve it with the arbitrary function h(x).
- Given the functiondetermine its Fourier transform. х — а х+а f(x) = rect + rect
- Graphically convolve, at least approximately, the two functions shown in Fig. P.11.31. Does that solution remind you of anything? Why is the convolution symmetrical? When does its peak value occur in
- Graphically convolve the two functions Æ’(x) and h(x) shown in Fig. P.11.26.Fig. P.11.26.How wide will the convolution be? Will it be symmetrical? Where will it start? h(x) f(x) 1 х +1 -2
- Figure P.11.25 depicts a single €œsaw tooth€ function and its convolution. Note that the convolution is asymmetrical€”explain why that€™s reasonable. Why does
- Given that ƒ(x) * h(x) = g(X), show that after shifting one of the functions an amount x0, we get ƒ(x - x0) * h(x) = g(X - x0).
- Referring to the previous problem, justify the fact that the convolution is zero for |X| ≥ d + ℓ when h is viewed as a spread function.
- Monochromatic plane waves perpendicularly illuminate a small circular hole in a screen. From point-P, beyond the hole on the central axis, exactly 3 Fresnel zones appear to fill the hole. If the
- Imagine a point source S a perpendicular distance Ï0from a circular hole in an aperture screen Î£. The screen is a distance r0in front of an axial observation point-P. Show
- Plane waves impinge perpendicularly on a screen with a small circular hole of radius R in it. It is found that when viewed from some axial point-P the hole uncovers 1/2 of the first half-period zone.
- Collimated light from a krypton ion laser at 568.19 nm impinges normally on a circular aperture. When viewed axially from a distance of 1.00 m, the hole uncovers the first half-period Fresnel zone.
- The circular hole in an opaque screen is 6.00 mm in diameter. It is perpendicularly illuminated by collimated light of wavelength 500 nm. How many Fresnel zones will be “seen” from a point-P on
- Derive Eq. (10.84). Em E1 |E| (10.84) 2
- Referring to Fig. 10.52, integrate the expression dS = 2Ï€Ï2sin Ï† dÏ† over the l th zone to get the area of that zone,Show that the mean distance to the lth
- Perform the necessary mathematical operations needed to arrive at Eq. (10.76). E = (-1)+1 2KƐspd (p + r) 2ΚκεΑρλ sin [wt – k(p + ro)] (10.76)
- Imagine an opaque screen containing 30 randomly located circular holes. The light source is such that every aperture is coherently illuminated by its own plane wave. Each wave in turn is completely
- What is the total number of lines a grating must have in order just to separate the sodium doublet (λ1 = 5895.9 Å, λ2 = 5890.0 Å) in the third order?
- A high-resolution grating 260 mm wide, with 300 lines per millimeter, at about 75° in autocollimation has a resolving power of just about 106 for λ = 500 nm. Find its free spectral range. How do
- A grating has a total width of 10.0 cm and contains 600 lines/mm. What is its resolving power in the second-order spectrum? At a mean wavelength of 540 nm, what wavelength difference can it resolve?
- Prove that the equationwhen applied to a transmission grating, is independent of the refractive index. a(sin 0)m — sin ө;) — т^ [10.61]
- Suppose that a grating spectrometer while in vacuum on Earth sends 500-nm light off at an angle of 20.0° in the first-order spectrum. By comparison, after landing on the planet Mongo, the same light
- Sunlight impinges on a transmission grating that is formed with 5000 lines per centimeter. Does the third-order spectrum overlap the second-order spectrum? Take red to be 780 nm and violet to be 390